On sets of extreme functions for Fatou's theorem

Bounded holomorphic functions on the disk have radial limits in almost every direction, as follows from Fatou’s theorem. Given a zero-measure set $E$ in the torus $\mathbb{T}$, we study the set of functions such that ${\lim }_{r\to 1^{-}}f(rw)$ fails to exist for every $w\in E$ (such functions were first constructed by Lusin). We show that the set of Lusin-type functions, for a fixed zero-measure set $E$, contain algebras of algebraic dimension $\mathfrak{c}$ (except for the zero function). When the set $E$ is countable, we show also in the several-variable case that the Lusin-type functions contain infinite dimensional Banach spaces and, moreover, contain plenty of $\mathfrak{c}$-dimensional algebras. We also address the question for functions on infinitely many variables.